Rotational Dynamics 1- Moments of Inertia and Equation of Motion

Workshop Tutorials for Biological and Environmental Physics MR9B: Rotational Dynamics I

A. Qualitative Questions:

1. When you’re sitting in a car which rounds a corner at high speed you can feel yourself pushed radially outwards against the car door or the person sitting next to you. If you were orbiting the Earth in the space shuttle you would not feel this force. Why is this the case?

2. Rebecca has gone to a conference in Cairns in northern Queensland, leaving Brent at home in Sydney to look after Barry the dog. a. Which one them has the greater angular speed, ? b. Which of them has the greater linear speed, v? Use a diagram showing their positions on the Earth to explain your answers.

B. Activity Questions:

1. Clocks What is the angular speed of the second hand? What are the angular speeds of the minute and hour hands? Does the size of the clock affect the angular speeds of the hands? Does it affect the linear speed of the ends of the hands?

2. Rotation platform What affects the 'slipping off' of the block? Which way does the block go as it slips off? Where is it more likely to slip, and why?

3. A Loaded Race Will all the cylinders roll down with same speed? Will all the spheres roll down with same speed? Try rolling them down the incline. Explain why some of them roll down faster than the others.

The Workshop Tutorial Project –MR9B: Rotational Dynamics I 117 C. Quantitative Question:

1. Many bacteria have flagella or cilia, tiny little waving appendages, which they use to propel them selves. These were always believed to just wave around to move the bacteria, but it turns out that some of them actually act as tiny propellers. However the smallest natural propeller is part of an ATPase molecule. An ATPase is an enzyme which either breaks down or builds up an ATP(Adenosine tri- phosphate) molecule. ATP is the energy currency of cells: energy is liberated by breaking one of the phosphate bonds, or stored by attaching a phosphate. The FI-ATPase molecule has seven sub-units, six of which form a ring around the seventh sub-unit, as shown below. This middle piece actually spins around like the rotor of an electric motor, but it was only by attaching another molecule like a propeller blade that it was possible to observe this movement. a. If each rotation takes 100 ms, what is the propeller angular velocity of the attached actin actin molecule molecule? b. If the actin molecule is 1 m long and has a mass of 2  10-22kg, what is the moment of inertia of the propeller? c. Assuming constant angular acceleration, if it takes 100 ms to perform a rotation starting FI-ATPase from rest, what is the angular acceleration of molecule the actin?

1 2 Irod = 3 ml .

2. Australia’s first wind farm is located on King Island, and the Tasmanian government is considering a plan to set up a new 130 MW wind farm in north west Tasmania. If you want to set up your own wind generator, it is possible to get plans from the internet to build your own home wind generator. One design uses a steel rotor which weighs around 20 kg, and is 2 m long.

pivot a. What is the moment of inertia of the rotor if you treat it as a single rod? The designer of this wind generator claims that it can produce 500 Watts in a 35 km.h-1 wind, which is not much more than a pleasant breeze. b. If the output from the generator is 15% of the kinetic energy of the rotor, how fast (how many revolutions per minute) must the rotor be doing to give this output? c. How many of these home made windmills would you need to set up a wind farm capable of producing 130 MW in a 35 km.h-1 wind?

1 2 Irod = 12 Ml . where l is the total length of the rod and M is the mass of the rod.

118 The Workshop Tutorial Project –MR9B: Rotational Dynamics I